Adaptive Wavelet Methods for Linear-Quadratic Elliptic Control Problems: Convergence Rates

We propose an adaptive algorithm based on wavelets for the fast numerical solution of control problems governed by elliptic boundary value problems with distributed or Neumann boundary control. A quadratic cost functional that may involve fractional Sobolev norms of the state and the control is to be minimized subject to linear constraints in weak form. Placing the problem into the framework of (biorthogonal) wavelets allows us to formulate the functional and the constraints equivalently in terms of $\ell_2$-norms of wavelet expansion coefficients and constraints in the form of an $\ell_2$ automorphism. The resulting first order necessary conditions are then derived as a (still infinite) system in $\ell_2$. Applying the machinery developed in [A. Cohen, W. Dahmen, and R. DeVore, Math. Comp., 70 (2001), pp. 27--75; A. Cohen, W. Dahmen, and R. DeVore, Found. Comput. Math., 2 (2002), pp. 203--245], we propose an adaptive method which can be interpreted as an inexact gradient descent method, where in each iteration step the primal and the adjoint system need to be solved up to a prescribed accuracy. Convergence of the adaptive algorithm is proved. In addition, we show that the adaptive algorithm is asymptotically optimal, that is, the convergence rate achieved for computing the solution up to a desired target tolerance is asymptotically the same as the wavelet-best N-term approximation of the solution, and the total computational work is proportional to the number of computational unknowns.